Bound states in Heisenberg antiferromagnets

Bound states in Heisenberg antiferromagnets

PHYSICS LETTERS Volume 30A, number 2 to the m a s s M o f the p a r t i c l e . In i ts n e i g h bourhood, the s p a t i a l g e o m e t r y i s bu...

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PHYSICS LETTERS

Volume 30A, number 2

to the m a s s M o f the p a r t i c l e . In i ts n e i g h bourhood, the s p a t i a l g e o m e t r y i s but s l i g h t l y m o d i f i e d by the e x i s t e n c e of the u n i v e r s e .

r,

BOUND

STATES

22 September 1969

References

1. Z.Hordk, Phys. Letters 28A (1968) 332. 2. C. Moller, The theory of relativity (Oxford 1955) p. 357.

IN HEISENBERG

ANTIFERROMAGNETS

S. W. LOVESEY and E. B A L C A R * Theoretical P h y s i c s Division, A . E. R . E. , H a r w e l l , B e r k s h i r e , UK

Received 2 August 1969

An equation is derived for the two-spin wave propagator for a Heisenberg antiferromagnet. The location of the bound state is given for a linear chain.

A f u n d a m e n t a l p r o b l e m in the study of H e i s e n b e r g a n t i f e r r o m a g n e t s is the d e t e r m i n a t i o n of the e x a c t g r o u n d s t a t e [2]. E x p e r i m e n t a l s t u d i e s of this p r o b l e m have not in the m a i n p r o v e d f r u i t ful b e c a u s e o t h e r s m a l l e f f e c t s m a s k that due to the d e v i a t i o n of the e x a c t ground s t a t e f r o m the N~el state. The p u r p o s e of this p a p e r i s to d r a w at t en t i o n to a d y n a m i c a l e f f e c t in H e i s e n b e r g a n t i f e r r o m a g n e t s that is a d i r e c t r e f l e c t i o n of this deviation. In the s p i n - w a v e a p p r o x i m a t i o n two spin wave m o d e s e a c h of e n e r g y ¢0q a r e found to e x i s t in a t w o - s u b l a t t i c e HeiseniSerg a n t i f e r r o m a g n e t (in the a b s e n c e of an e x t e r n a l m a g n e t i c field); the e x c i t a t i o n of an a - t y p e s p i n - w a v e c a u s e s the t o t a l z - c o m p o n e n t of spin to be d e c r e a s e d by one unit wh i l e the e x c i t a t i o n of a fl-type mode c a u s e s it to be i n c r e a e d by one unit. The ~ and fl o p e r a t o r s and the c o r r e s p o n d i n g t r a n s f o r m a t i o n c o e f f i c i e n t s u q and V q , a r e defined in a c c o r d with the notation of K i t t e l [3]. In t e r m s of this t h e o r y the longitudinal c o m p o n e n t of the i n e l a s t i c m a g n e t i c c r o s s s e c t i o n f o r the s c a t t e r i n g of t h e r m a l n e u t r o n s [4] with i n c i d e n t wave v e c t o r k and s c a t t e r e d wave v e c t o r k ' can be w r i t t e n d2~ ~ ~ D 8 5 x d~lEe ~" 1234 1+3,2+4 K+%1-2 A(1 2 3 4 ) ×

(-2) ImG(12 34;

2.o,

z x 1.5.

TWO- MAGNON

BAND

÷

/

a

/

~-"~1.0.

/

S=~-1

0.5

¸

(I) ¢o), 10 o

* On leave from the Austrian Study Group for Atomic Energy Physics Department, Reactor Center, Seibersdorf, Austria. 84

6o° Fig. 1.

9'o"

Volume 30A. number 2

P HYSI C S L E T T E R S

A (1334) -- {u lv2[v 3 u4 - u3v 4 exp(i~ • r)]} + (u = v } * . (2) In (1), 1 =- q l e t c . , r a r e the r e c i p r o c a l v e c t o r s of a s u b l a t t i c e , ~J the v e c t o r j o i n i n g n e a r e s t n e i g h b o u r s on o p p o s i t e s u b l a t t i c e s , x = k-k', a n d co = (k2-k'2)/2n~ T h e t w o - t i m e G r e e n f u n c t i o n G i s d e f i n e d a s [7].

On s u b s t i t u t i n g G d e r i v e d in the l i n e a r s p i n w a v e a p p r o x i m a t i o n and a t a b s o l u t e z e r o in (1) the c r o s s s e c t i o n i s s e e n to b e n o n z e r o only if co = Wl + co2 and K+ ~ = 1-2. C o w l e y e t al. [5] h a v e r e p o r t e d the o b s e r v a t i o n of t h i s d i f f u s e t w o m a g n o n s c a t t e r i n g in C o F 2. If i n s t e a d of p r o p a g a t i n g i n d e p e n d e n t l y of one a n o t h e r the a and fl t y p e s p i n w a v e s i n t e r a c t and f o r m a b o u n d s t a t e of w a v e v e c t o r K a n d e n e r g y o~ , t h e l a t t e r w i l l m a n i f e s t i t s e l f in the c r o s s B section as a uniquely defined excitation with K+T = K a n d ¢o= coB. T h e e x i s t e n c e of the b o u n d state exditation atzero wave vector has been r e p o r t e d w i t h r e c e n t o p t i c a l e x p e r i m e n t s [1]. We g i v e a t h e o r y of t h e d i s p e r s i o n of the e x c i t a t i o n together with numerical results for a oneodimen s i o n a l chain. U t i l i z i n g the D y s o n - M a l e e v t r a n s f o r m a t i o n w e f i n d in t h e H a r t r e e - F o c k a p p r o x i m a t i o n the f o l l o w i n g e q u a t i o n f o r t h e G r e e n f u n c t i o n (3) at absolute zero :

{co - (l÷eo/2rS)(cok+coq)}G(k q k'q'; co) = 1 =~6k'k'Sq'q'

2rJ ~,2 ---N- 1 5k_q,l_2G(12k'q';

(4)

w)×

1,'1, ~.(k,q)~ ,

V~:~(k,q) =

22 September 1969

UlV2{UqVk~k_ q + UqUk~q}+

(5)

+ UlU2{UkUq~k. 1 + UqVk~l~. H e r e , S i s t h e s p i n of the m a g n e t i c i o n s , l+eJ2rS the H a r t r e e , F o c k r e n o r m a l i z a t i o n factor~ [2], J and rk a r e d e f i n e d a s u s u a l [3]. T h e s a m e s c h e m e u s e d to d e r i v e eq. (4) g i v e s the e x a c t c o n d i t i o n f o r the l o c a t i o n of the b o u n d s t a t e in a H e i s e n b e r g f e r r o m a g n e t [6]. T h e s o l i d c u r v e s in fig. 1 g i v e s the l o c a t i o n of the p o l e s of the G r e e n f u n c t i o n (4) f o r the p a r t i c u l a r c a s e of a l i n e a r chain. T h e b r o k e n c u r v e m a r k s a d i s c o n t i n u i t y in the G r e e n f u n c tion due to a s i n g u l a r i t y in the d e n s i t y of s t a t e s . Note t h a t a l l m o d e s l i e i n s i d e the t w o - m a g n o n c o n t i n u u m and t h e r f o r e h a v e a w i d t h a s s o c i a t e d with t h e m . A d e t a i l e d a n a l y s i s of the c r o s s s e c tion f o r R b M n F 3 t y p e c r y s t a l s w i l l a p p e a r e l s e where. T h e a u t h o r s a r e i n d e b t e d to Dr. J. H u b b a r d f o r s e v e r a l s t i m u l a t i n g d i s c u s s i o n s on the s u b j e c t of the p a p e r .

References 1. R . J . Elliott, M . F . Thorpe, G. Imbusch, R. London and J. B. Parkinson, Phys. Rev. Letters 21 (1968) 147. 2. F.Keffer, Handbuch der Physik 18 (Springer) § 43. 3. C.Kittel, Quantum Theory of Solids (John Wiley, 1965) ch. 4. 4. L. Van Hove, Phys. Rev. 95 (1954) 249. 5. R.A. Cowley, W . J . L . Buyers. P. Martel and R. W. H.Stevenson, Phys. Rev. Letter 23 (1969) 86. 6. M. Wortis, Phys. Rev. 132 (1963) 85. 7. D.N. Zubarev, Soy. Physics -Uspekhi 3 (1960) 320.

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